An Introduction to Derivatives

Date: 05/17/98 at 14:15:43
From: Erika
Subject: Third derivative?
Hello, I have to research derivatives for a project. It asks about a
third derivative. I haven't seen a second mentioned so I was
wondering, is there a third derivative?
Thank you for your time.

Date: 05/20/98 at 22:44:11
From: Doctor Pat
Subject: Re: Third derivative?
Erika,
Yes, there is a second derivative, and a third, and a fourth... at
least as long as the function continues to be differentiable. With
polynomial functions you can take derivatives forever, but after long
enough they all get kind of boring. I'll let you figure out why.
A derivative is just the rate of change of one thing as measured by
the change in another.
Wow, I find that hard to understand and I wrote it. How about we look
at a real world example?
If you go from here to there, you change your position. Let's pretend
there is a number line along which you are walking, and a big clock on
the wall. As you walk, your position and the clock are both changing.
Your velocity (speed with a direction attached, but you knew that
already, I suspect), which is the First derivative of position with
respect to time, dP/dt, is how far you go divided by how long it
takes. You may recognize this as the sixth grade formula for rate:
distance
rate = --------
time
In calculus, we make the change in time smaller and smaller and use
limits to come up with an "instantaneous" velocity. If you are moving
with a constant speed the average speed (sixth grade method) and the
dP/dt (really cool calculus method) are the same.
So what is a second derivative? Well if you were walking along and
decided to walk a little faster, the first derivative (your velocity)
would also increase. The second derivative measures the change in the
first derivative per unit of "whatever" (in our example, time).
The common language for the change in velocity per unit of time is
acceleration. When you step on the gas, you "change" your speed
(velocity). How fast it is changing is the idea of a second
derivative. And if your second derivative is changing, guess what we
call that? We don't have names for all the different derivatives, or
at least I don't know the names if they do exist, but you can see how
each one is related to the one before it. Graphically you should also
try to understand that if you graph y versus the nth derivative of x,
then the slope of the curve at any point is the (n+1)th derivative of
y with respect to x.
Let's walk through the example with some computations of derivatives.
If you do not know how to calculate derivatives and are not expected
to, ignore all of this.
If the position of some object is given as x(t) = t^3 + 2t + 1, then
we can figure its position at any time by evaluating for the value
of t. At t = 0, the position is x = 1 at t = 1 the position is x = 4.
We don't know how it got there, or where else it has been, but we know
that between t = 0 and t = 1 it moved 3 units to the right. Its
AVERAGE velocity was +3 units per time unit. If we take the derivative
of x(t) with respect to time, we get the velocity. This is the FIRST
derivative:
x'(t) = 3t^2 + 2
We can use this function the same way we used the position function,
only we get the velocity at each moment. For example at time t = 0 the
velocity was 2, and at time t = 1 the velocity was 5. From this we can
see that not only is it moving to the right, but it is speeding up.
We can expect that it will move even farther to the right in the next
unit of time than it did before.
As before, now that we know two values of a function, we can figure
the average acceleration (change in velocity/change in time) and get
3 units per time unit squared. If we want the instanteous acceleration
at any moment, we take the derivative of velocity, which is the second
derivative of position:
x''(t) = v'(t) = a(t) = 6t
This is a function which gives us the acceleration at any time. By now
I think you get the picture. By the way, the third derivative of
position with respect to time, which is the change in acceleration
with respect to time, is named for what happens when you experience a
sudden change of acceleration: you feel a jerk.
Hope that helps. There are lots of ideas and lots of language involved
in the ideas of calculus, but they really help to explain ideas that
involve change and motion in a way earlier math could not. That is why
calculus was so necessary for Kepler and Newton when they worked with
the orbits of planets and the laws of gravity.
If you need help on the mechanics of calculating a derivative, drop us
another note.
-Doctor Pat, The Math Forum
Check out our web site! http://mathforum.org/dr.math/

Date: 05/22/98 at 08:19:42
From: Goodeffort
Subject: Re: Third derivative?
Thank you for helping me. The information you sent was just what I
needed! :-)